Optimal. Leaf size=158 \[ \frac {2 a \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}}+\frac {32 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac {64 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {32 a \sin (c+d x)}{45 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.24, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2770, 2759, 2751, 2646} \[ \frac {2 a \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}}+\frac {32 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac {64 \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {32 a \sin (c+d x)}{45 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2759
Rule 2770
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \, dx &=\frac {2 a \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {8}{9} \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {16 a \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {16}{21} \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {16 a \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {32 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d}+\frac {32 \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{105 a}\\ &=\frac {16 a \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {64 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {32 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d}+\frac {16}{45} \int \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {32 a \sin (c+d x)}{45 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {64 \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {32 (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{105 a d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 92, normalized size = 0.58 \[ \frac {\left (1890 \sin \left (\frac {1}{2} (c+d x)\right )+420 \sin \left (\frac {3}{2} (c+d x)\right )+252 \sin \left (\frac {5}{2} (c+d x)\right )+45 \sin \left (\frac {7}{2} (c+d x)\right )+35 \sin \left (\frac {9}{2} (c+d x)\right )\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)}}{2520 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 72, normalized size = 0.46 \[ \frac {2 \, {\left (35 \, \cos \left (d x + c\right )^{4} + 40 \, \cos \left (d x + c\right )^{3} + 48 \, \cos \left (d x + c\right )^{2} + 64 \, \cos \left (d x + c\right ) + 128\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 129, normalized size = 0.82 \[ \frac {1}{2520} \, \sqrt {2} \sqrt {a} {\left (\frac {35 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {45 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {252 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {420 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d} + \frac {1890 \, \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 97, normalized size = 0.61 \[ \frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (560 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-800 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+552 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-104 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+107\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 79, normalized size = 0.50 \[ \frac {{\left (35 \, \sqrt {2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, \sqrt {2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 252 \, \sqrt {2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 420 \, \sqrt {2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1890 \, \sqrt {2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\cos \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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